Generalized iterated-sums signatures

We explore the algebraic properties of a generalized version of the iterated-sums signature, inspired by previous work of F.~Kir\'aly and H.~Oberhauser. In particular, we show how to recover the character property of the associated linear map over the tensor algebra by considering a deformed quasi-shuffle product of words on the latter. We introduce three non-linear transformations on iterated-sums signatures, close in spirit to Machine Learning applications, and show some of their properties.


Introduction
Recently, a series of papers [ DEFT20a , DEFT20b , KO19 , LO17 ] have highlighted the importance of signature-like objects (following T. Lyons' nomenclature) for capturing features of sequentially ordered data.Part of the reason for their success is that these transformations posses a universality property, meaning that they are able to approximate arbitrary1 nonlinear mappings on sequence space by linear functionals on feature space.They can also be efficiently computed thanks to an inherent recursive structure.
Both properties can be succinctly described by using Hopf-algebraic language, which has by now become standard in the field.For the so-called iterated-integrals signature, the underlying Hopf algebra is the space of words together with the commutative shuffle product [ Ree58 ,Reu93 ] and the noncocommutative deconcatenation coproduct.On the other hand, replacing integrals by sums, we obtain the iterated-sums signature, which is defined over the commutative quasi-shuffle algebra on words [ Car72 ,Gai94 ,Hof00 ,NR79 ].Equipped with the aforementioned deconcatenation coproduct, the latter becomes a Hopf algebra.Shuffle and quasi-shuffle products algebraically encode integration and summation by parts for iterated integrals and sums respectively.In both cases the properties mentioned in the preceding paragraph amount to saying that the corresponding signature-like maps are characters, i.e., algebra morphisms, and that they satisfy Chen's relation.
Recently, F. Király and H. Oberhauser [ KO19 ] introduced a higher order version of the iterated-sums signature as a way of approximating the iterated-integrals signature.The main disadvantage of this generalization is that the character property is lost and consequently the universality property ceases to hold.In the paper at hand we unfold the algebraic underpinning of the definition of Király and Oberhauser's higher order iterated-sums signature, which permits further generalization to arbitrary nonlinearities, as opposed to the more restricted class of exponential-type nonlinearities underlying previous approaches.Moreover, we show that the generalized iterated-sums signature enjoys a character property with respect to a different Hopf algebra defined on words in terms of a modified quasi-shuffle product and the deconcatenation coproduct; in fact, we show that the algebraic structure actually depends on the selected nonlinear transformation.We come back to this topic in Section 3.1.1 .
Thanks to the more general approach, we are able to introduce three new transformations of the iterated-sums signature.The first transformation is obtained by applying a tensorized nonlinear transformation to each time slice, the second one is constructed by applying a polynomial map to increments, whereas the third is obtained by first transforming the data and then considering its increments.We show that these transformations can be expressed in terms of the un-transformed iterated-sums signature.In the third case, this rewriting procedure generalizes earlier work by L. Colmenarejo and R. Preiß on iterated-integrals signatures defined with respect to paths transformed by polynomial maps [ CP20 ].
The rest of the paper is organized as follows.In Section 2 , we review the algebraic foundation of our construction, i.e., the notion of quasi-shuffle Hopf algebra.In Section 3 , we introduce the generalized iterated-sums signature map and provide a complete description of its most important properties using the developments of the previous section.

Acknowledgments:
We thank the referee for pertinent remarks and observations improving our understanding of certain algebraic aspects, which ameliorated the presentation.We also thank Rosa Preiss for helpful comments.The second author is supported by the Research Council of Norway through project 302831 "Computational Dynamics and Stochastics on Manifolds" (CODYSMA).The third author is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy -The Berlin Mathematics Research Center MATH+ (EXC-2046/1, project ID: 390685689)

Quasi-shuffle Hopf algebra
In this section we briefly recall the notion of commutative quasi-shuffle product, the algebraic framework present in [ DEFT20a ].However, we shall emphasise the refined viewpoint based on the notion of half-shuffle product.Readers interested in the details are directed to further references [ EFP20 , Hof00 , FP20 , FPT16 , HI17 ].
Following Foissy and Patras [ FP20 ] we define the notion of commutative quasi-shuffle algebra over the base field R. Definition 2.1.A commutative quasi-shuffle algebra pA, ą, ‚q consists of a nonunital commutative R-algebra pA, ‚q equipped with a linear half-shuffle product ą : A b A Ñ A satisfying x ą py ą zq " px ˚yq ą z, (1) where the quasi-shuffle product ˚: A b A Ñ A is defined as One verifies that relations ( 1 ), ( 2) and ( 3 ) imply that the quasi-shuffle product ( 4 ) is both commutative and associative.Observe that one may characterise a commutative quasishuffle algebra as a space with two commutative products related through a -symmetrizedhalf-shuffle product.In the following, quasi-shuffle algebra means commutative quasi-shuffle algebra.
Any quasi-shuffle morphism is an algebra morphism between quasi-shuffle algebras, that is, Λpx ˚yq " Λpxq ˜Λpyq.A quasi-shuffle algebra pA, ą, ‚q has a unital extension.Indeed, set Ā -R1 ' A and define for a P A: 1 ‚ a " a ‚ 1 -0, 1 ą aa and a ą 1 -0.Note that the product 1 ą 1 as well as 1 ‚ 1 are excluded and so the identity 1 ˚1 -1 must be imposed in addition.This turns Ā into a unital algebra with quasi-shuffle product ˚.Furthermore, any quasi-shuffle morphism can be extended as a unital algebra morphism between the corresponding unitizations; in particular, this extension will preserve the units.
If the commutative algebra pA, ‚q in Definition 2.1 has a trivial product, i.e., x ‚ y " 0 for all x, y P A, then the notion of commutative quasi-shuffle algebra reduces to that of commutative shuffle algebra, which is defined solely in terms of relation ( 1 ).In this case the commutative and associative product ( 4 ) is called shuffle product.
Remark 2.3.We note that different terminologies are present in the literature.Commutative shuffle and quasi-shuffle algebras are also known as Zinbiel and commutative tridendriform algebras, respectively.The noncommutative generalizations of both shuffle and quasi-shuffle algebra, are also known as dendriform and tridendriform algebras, respectively.We follow [ FP20 ], where the preference for the terminology used in this work is explained.
Our main example provides also the paradigm of commutative quasi-shuffle algebra, i.e., the free commutative quasi-shuffle algebra.Let A " t1, . . ., du be a finite alphabet and consider the reduced symmetric algebra SpAq over the vector space spanned by it.By way of explanation, SpAq is the space spanned by words in commuting letters from A; here reduced means that we do not suppose SpAq to have a unit, i.e., we consider only the augmentation ideal of the standard symmetric algebra over A. Keeping up with our previous convention [ DEFT20a ], we denote the commutative product in SpAq by square brackets.Neither endow we SpAq with any additional algebraic structure other than its product.Finally we recall that SpAq has a natural grading SpAq " where S n A is spanned by products of the form ri 1 ¨¨¨i n s with i 1 , . . ., i n P A. We denote this basis by A n .It is well known that dim S n A " `d`n´1 n ˘.Furthermore, the set A n constitutes a basis for SpAq.Now, we let H -T pSpAqq be the unital tensor algebra over SpAq.As a vector space, it is the direct sum where H 0 " Re and We also set In the following we will use the word notation for elements in H.For example, when d " 2, a generic element of H might look like Concatenation, written by juxtaposition of symbols, is the standard product on H.In particular, H inherits a grading from SpAq in this way, which we call the weight and denote by | ¨|.The length of a word w " s 1 ¨¨¨s k P H is defined to be pwq " k.In [ DEFT20a ] we show that It can also be shown that the dimensions satisfy the following recursion: The standard basis for H is the set of words over A, here denoted by A ˚. We endow H with an inductively defined product obtained from the bracket product of SpAq: e ‹ uuu ‹ e and ua ‹ vb " pu ‹ vbqa `pua ‹ vqb `pu ‹ vqrabs (5) for u, v P H and a, b P SpAq.Hoffman [ Hof00 ] called ( 5 ) quasi-shuffle product and showed that it is commutative and associative as well as compatible with the deconcatenation coproduct ∆ dec : H Ñ H b H, defined on basis elements u " u 1 ¨¨¨u n P A ˚by and the counit ε determined by the grading, so that H qsh -pH, ‹, ∆ dec , e, ε, αq is a Hopf algebra.See also [ NR79 ] and [ Gai94 ].
The non-unital quasi-shuffle algebra H qsh carries a commutative quasi-shuffle structure [ Lod07 ], defined recursively by u ą va :" pu ‹ vqa, ua ‚ vb " pu ‹ vqrabs. (6) It is such that the unitization of H qsh is isomorphic to H qsh as a unital commutative algebra.Observe that any word w " s 1 ¨¨¨s k P H can be written using the half-shuffle product defined in ( 6 ) w " p¨¨¨pps 1 ą s 2 q ą s 3 q ¨¨¨q ą s k .
Remark 2.4.It is natural to consider the relation between the deconcatenation coproduct ∆ dec and half-shuffle as well as the ‚ products.It turns out that they form what is known as quasi-shuffle bialgebra.See reference [ FP20 ] for more details.
Finally, we recall the following important result due to Loday [ Lod07 , Theorem 2.5].
Theorem 2.5.The free commutative unital quasi-shuffle algebra over R d is isomorphic to H qsh .
The (algebraic) dual space H 1 qsh can be identified with formal word series T " ÿ wPA ˚xT, wyw, in the sense that there is an isomorphism between such formal series and elements of H 1 .The convolution product of two maps R, T P H 1 is defined by Observe that this product is associative but not commutative.
2.1.Deformed quasi-shuffle products.In this section, we describe how coalgebra automorphisms of the coalgebra pH, ∆ dec q can be used to define new quasi-shuffle structures on H by deforming Hoffman's quasi-shuffle product.We recall that a coalgebra morphism between two coalgebras pA, ∆ A , ε A q and pB, ∆ B , ε B q is a linear map Ψ : Remark 2.6.All possible coalgebra endomorphisms of pH, ∆ dec q have been characterized by Foissy, Patras and Thibon [ FPT16 , Corollary 2.1].In our context, the result is that this class is isomorphic to the class of linear maps from H `to SpAq.The isomorphism is described in the following way: any linear map ζ : H `Ñ SpAq induces a unique coalgebra endomorphism Conversely, any coalgebra endomorphism ϕ : pH, ∆ dec q Ñ pH, ∆ dec q is uniquely determined by the knowledge of ζ " π ˝ϕ, where π denotes the projection of H onto SpAq, which is orthogonal to R and SpAq bk for k ě 2. In general, the result remains valid if SpAq is replaced by any commutative algebra.Moreover, in [ FPT16 ] it is also shown that ϕ is an automorphism if and only if the restriction of ζ " π ˝ϕ to SpAq is a linear isomorphism.
From now on, Ψ will denote a coalgebra automorphism of pH, ∆ dec q.The following three assertions can be derived straightforwardly by transporting all algebraic structure through the mapping Ψ.
Proposition 2.7.The space H `equipped with the deformed products is associative and commutative.
(1) It is well known that when Ψ is Hoffman's exponential, the deformed quasi-shuffle product ‹ Ψ coincides with the classical shuffle product on H.We note, however, that the splitting given in Equation (7) does not coincide with its standard half-shuffle decomposition; in particular, the associative product ‚ Ψ is non-zero.This is consistent with the fact, noted in [ DEFT20a , Remark 5.10], that Hoffman's exponential is not a half-shuffle morphism.
(2) Defining the area 2 operations area Ψ pu, vqu ą Ψ v ´v ą Ψ u and areapu, vqu ą v ´v ą u, for u, v P H, we see immediately that This is an isomorphism property of Tortkara algebras, introduced by Dzhumadil'daev in 2007 [ Dzh07 ].
2.2.Coalgebra morphisms induced by formal power series.We now introduce a special class of coalgebra automorphisms of H qsh , described in [ Hof00 ,HI17 ].Recall that composition of an integer n ě 1 refers to a sequence I " pi 1 , . . ., i k q of positive integers such that i 1 `¨¨¨`i k " n.We write Cpnq for the set of all compositions of n.For any word w " s 1 ¨¨¨s n P H and composition I " pi 1 , . . ., i k q P Cpnq we define the word Irws P H Irws -rs 1 ¨¨¨s i1 srs i1`1 ¨¨¨s i1`i2 s ¨¨¨rs i1`¨¨¨`i k´1 `1 ¨¨¨s n s.
Formal diffeomorphisms f P tRrrtss induce linear endomorphisms of H in the following way: suppose that and define Remark 2.11.Note that this a special case of Remark 2.6 with ζpa 1 ¨¨¨a k q :" c k ra 1 ¨¨¨a k s.
Formal diffeomorphisms with c 1 ‰ 0 are invertible with respect to the composition of formal power series.In this case, Ψ f is an automorphism, and it can be shown that Ψ f ˝Ψg " Ψ f ˝g , and in particular Ψ ´1 f " Ψ f ´1 , [ HI17 ].Finally, we observe that Ψ f is always a coalgebra morphism [ FPT16 ]; this means that the identity ∆ dec ˝Ψf " pΨ f b Ψ f q ˝∆dec holds.Moreover, by definition, the map Ψ f is graded, that is, Ψ f pH n q Ă H n for all n ě 0. Therefore, any invertible diffeomorphism induces a deformed quasi-shuffle algebra which we denote by H f .
In Section 3.1 we restrict ourselves to deformed quasi-shuffles induced from invertible formal diffeomorphisms, since in this case the map Ψ f has a direct interpretation in terms of the iterated-sums signature.

Iterated-sums signatures
We start this section by recalling the definition of the iterated-sums signature introduced in [ DEFT20a ].Fix integers d ě 1 and N ą 0. A d-dimensional time series of length N is a sequence of vectors x " px 0 , . . ., x N ´1q P pR d q N .The following notation for elements in the time series x is put in place: x j " px p1q j , . . ., x pdq j q, and it is extended to include brackets in A by defining 2 This terminology comes from the interpretation of half-shuffles as integration operators (see e.g. [ DEFT20a , Section 5.1]) Given a d-dimensional time series of length N , its increment series, denoted by δx, is also a d-dimensional time series of length N ´1 with entries defined by δx k -x k ´xk´1 .We also denote, for a " ri 1 ¨¨¨i n s P A, Definition 3.1.Let x be a d-dimensional time series, and denote by δx its increment series.
The iterated-sums signature of x is the two-parameter family pISSpxq n,m : 0 ď n ď m ď N q of linear maps in H 1 qsh such that ISSpxq n,n " ε, and defined recursively by xISSpxq n,m , ey -1, and for all words a 1 ¨¨¨a p P A ˚.
We recall that, as a formal word series, the map ISSpxq n,m can be expressed as the time-ordered product (defined with respect to the concatenation product in T pSpAqq) In fact, eq. ( 10) can be seen to arise as the solution to a fixed-point equation in H qsh .Indeed, from Definition 3.1 we see that for any word a 1 ¨¨¨a p P A ˚, δxISSpxq n,¨, a 1 ¨¨¨a p y m " xISSpxq n,m , a 1 ¨¨¨a p y ´xISSpxq n,m´1 , a 1 ¨¨¨a p y " xISSpxq n,m´1 , a 1 ¨¨¨a p´1 yδx ap m .Therefore, the equality between word series δ ISSpxq n,m " ISSpxq n,m´1 Φpδx m q, ISSpxq n,n " ε holds, where the "polynomial extension" map Φ : R d Ñ SppAqq (c.f.[ NT10 , eq. 60]) is defined by Φpzq " 8 ÿ n"1 ˜ÿ iPA z piq ris ¸n " ÿ aPA z a a, (with respect to the commutative product in SppAqq).Note that Φ amounts to a geometric series in the completed symmetric algebra SppAqq -recall from Section 2 that the bracket r¨¨s denotes the symmetric tensor product in SpAq.This extension has also been considered in a Machine Learning context by Toth, Bonnier and Oberhauser [ TBO20 ].
We now record the two most relevant properties of the iterated-sums signature of a ddimensional time series, shown in [ DEFT20a , Theorem 3.4].Theorem 3.2.
(1) For each 0 ď n ď m ď N , the map ISSpxq n,m is a quasi-shuffle algebra character.
In fact, in light of the commutative quasi-shuffle structure of H qsh , one can be more precise about the nature of item (1) in Theorem 3.2 .Let X N denote the space of real-valued time series with fixed time horizon N P N. It carries itself a commutative quasi-shuffle structure, given by The corresponding commutative and associative quasi-shuffle product is px, yq Þ Ñ px ¨x 0 qpy ¨´y 0 q in X N .For a given d-dimensional time series x, we define a map σpxq By Theorem 2.5 , it admits a unique extension to H qsh as a commutative quasi-shuffle morphism (in the sense of Definition 2.2 ).
Proposition 3.4.The unique extension σpxq : H qsh Ñ X N is such that for all 0 ď k ď N and words w P A ˚we have xσpxq, wy k " xISSpxq 0,k , wy.
Proof.We first observe that since ri 1 s ‚ ¨¨¨‚ ri n s " ri 1 ¨¨¨i n s for all i 1 , . . ., i n P A, we have xσpxq, ri 1 ¨¨¨i n sy k " pxσpxq, ri 1 sy d ¨¨¨d xσpxq, ri n syq k " xISSpxq 0,k , ri 1 ¨¨¨i n sy by eq. ( 9) .This shows the identity for all words of length 1.Now, suppose the equality is proven for all words up to length p.Any word w of length p `1 can be decomposed as w " ua for some u P A ˚with puq " p and a P A. Since, from eq. ( 6) (set v " e), u ą a " ua for any u P A ˚and a P A, we see that To construct the maps x Þ Ñ ISSpxq n,k for 0 ď n ď k ď N , one can follow a similar route, by first considering the map x Þ Ñ x " px k : n ď k ď N q with xk " x k ´xn .The image of X N under this map will be denoted by Xn,N .The quasi-shuffle structure defined above can be transported to Xn,N via this map, i.e., x ľ ỹ -Č x ľ y, x d ỹ :" Ć x d y.It is not difficult to see that then the same procedure applied now to Xn,N gives rise to ISSpxq n,k for 0 ď n ď k ă N .In particular we obtain the following hold for all u, v P H qsh .
Remark 3.6.In more concrete terms, the content of Proposition 3.5 can be interpreted as computation formulas for the entries of ISS.Indeed, since every symbol a P A which is not a single letter can be written as a " a 1 ‚ i for some a 1 P A with |a 1 | " |a| ´1 and i P A, entries indexed by A can be computed inductively using eq.(12) , starting from single letters.Next, one notes that every non-empty word w P A ˚can be decomposed as w " w 1 a " w 1 ą a for some w 1 P A ˚and a P A with |w| " |w 1 | `|a|, entries indexed by words with length greater than 1 can be computed inductively using eq.( 11) and the values computed in the previous step.
In addition to the map Ψ f described in Section 2.2 , it induces a transformation on formal word series by Definition 3.7.Let x be a d-dimensional time series and f P tRrrtss a formal diffeomorphism.
The generalized iterated-sums signature is the family of linear maps pISS f pxq n,m : 0 ď n ď m ď N q defined by ISS f pxq n,m -ź năjďm ˜ε `fb ˜ÿ aPA δx a j a ¸¸.

We immediately have
Proposition 3.8.The generalized iterated-sums signature satisfies Chen's property, that is, for any We observe that due to the nonlinear nature of the transformation f applied inside the product, expansion of this expression as a proper word series is, in principle, not straightforward.However we have the following result.Proposition 3.9.For every w P H, where Ψ f is defined in eq.(8) .
3.1.1.Relation to F. Király and H. Oberhauser.We relate our results to the higher-order discrete signatures introduced by F. Király Here x i " ř jPA δx pjq i rjs and px i q bn " ř j1,...,jnPA δx pj1q i ¨¨¨δx pjnq i rj 1 s ¨¨¨rj n s.We note that this map, considered as a linear map on T pR d q, is not an algebra morphism over any product defined on T pR d q that is compatible with the grading.Indeed, suppose there is such a product and denote it by f.Consider the map S ppq pxq over a single time step with a non-zero increment.Fix moreover a single symbol, say 1 P A. Then pδx p1q 0 q p`1 " xr1 fpp`1q s, S ppq pxqy " 0 which is a contradiction.This can be resolved by considering the infinite-dimensional polynomial extension of the time series, including all powers of increments [ TBO20 , Example B.2].In essence, this is what the quasi-shuffle approach does -the extension is obtained by considering the bracket terms ri 1 ¨¨¨i n s P A and the corresponding extended increments δx ri1¨¨¨ins .
However, even when considering the proper extension, the map so obtained does not yield a character over the quasi-shuffle Hopf algebra when p ą 1 (the case p " 1 corresponds to ISSpxq, see eq. ( 10) ).Taking p " 2, a single time step and considering the product r1s ‹ r1s constitutes a simple counterexample. 3In this case, the analogue of eq. ( 13) consider the 3 The reader is invited to work out the details.
full extension equals ISS fp pxq, with f p " t `1 2 t 2 `¨¨¨`1 p! t p ; Corollary 3.10 restores the character property of this map, with respect to a different product.Moreover, we have that S ppq pxq " πpISS fp pxqq where π : H 1 Ñ H 1 1 is the unique concatenation morphism satisfying πprisq " ris and πpri 1 ¨¨¨i n sq " 0 if n ą 1.
Finally, we mention that in the limit p Ñ 8, S p8q pxq coincides with the iterated-integrals signature of the path X interpolating the values of x piecewise linearly with unit speed.In the same way, the extended version ISS f8 pxq coincides with the iterated-integrals signature of an infinite-dimensional extension of X [ DEFT20a , Theorem 5.3].Both statements are consistent with the fact that f 8 ptq " expptq ´1, so that Ψ f8 is the Hoffman exponential and ‹ f8 becomes the shuffle product (over A and A, respectively).

Polynomial transformation of the time series' increments.
In some applications, one might have only access to observables of a time series and not to the time series itself.In others, such as in Machine Learning, applying nonlinearities to the data might be of use.We introduce now an analogue of the iterated-sums signature, acting on transformed data.In the following, we will work with different base alphabets, so we explicitly include the size of it in the notation.Hence from now on we write e.g.H qsh pR d q to indicate this.
Let P : R d Ñ R e be a polynomial with P p0q " 0, and fix a d-dimensional time series x.We are interested in describing the algebraic properties of the iterated-sums signature of the transformed increments pP pδx 0 q, . . ., P pδx N ´1qq.That is, we wish to study the map xISS P pincq pxq n,m , a 1 ¨¨¨a p y :" It is immediate from the definition that, as a word series, ISS P pincq n,m pxq admits the factorization In particular we have Proposition 3.14.For 0 ď n ď n 1 ď n 2 ď N , the identity ISS P pincq pxq n,n 1 ISS P pincq pxq n 1 ,n 2 " ISS P pincq pxq n,n 2 holds.
Since P vanishes at 0, the entries of ISS P pincq pxq are invariant to time-warping.Therefore, since the iterated-sums signature contains all such invariants, we are guaranteed to be able to express all said entries in terms of those in ISSpxq.In order to describe this relation, we consider a map Φ P : HpR e q Ñ HpR d q induced by P .Recall that for a multi-index ν " pν 1 , . . ., ν d q P N d 0 , and x " px p1q , . . ., x pdq q P R d one writes |ν| -ν 1 `¨¨¨`ν d and x ν -px p1q q ν1 ¨¨¨px pdq q ν d .Now, expressing each component of P " pp 1 , . . ., p e q as p j pxq " we set p ˛pjq -ÿ |ν|ďdeg pj p j;ν r1 ν1 ¨¨¨d ν d s for all j P B " t1, . . ., eu.Note that this expression can be more succinctly written as with ν j pri 1 ¨¨¨i n sq " #tk : i k " ju.This map extends uniquely to a morphism of commutative algebras, p ˛: SpBq Ñ SpAq.It then has a unique extension Φ P to all of H as a concatenation morphism, i.e. if w " b 1 ¨¨¨b m P B ˚then Φ P pwq " p ˛pb 1 q ¨¨¨p ˛pb m q.
Lemma 3.16.The map Φ P : H qsh pR e q Ñ H qsh pR d q is a quasi-shuffle morphism in the sense of Definition 2.2 .Moreover, it is a morphism of Hopf algebras.
Proof.By definition, Φ P preserves the ‚ product in H qsh .Since it is a concatenation morphism, it must also preserve the half-shuffle ą.Indeed, for u P B ˚and b P B we have that Φ P pu ą bq " Φ P pubq " Φ P puqΦ P pbq " Φ P puq ą Φ P pbq.
Thus, Φ P is a quasi-shuffle morphism, and in particular an algebra morphism.
In order to check that it is also a coalgebra morphism, it suffices to prove the property for j P B. In this case we have that ∆ dec Φ P prjsq " Remark 3.17.Observe that for P : R e Ñ R f and Q : R d Ñ R e the composition rule Φ P ˝Q " Φ Q ˝ΦP holds.Indeed, for a single letter j P C -t1, . . ., pu we see that (using the notation introducing in eq. ( 16) ) p j;νpaq rq ˛pi 1 q ¨¨¨q ˛pi n qs " ÿ a"ri1¨¨¨insPA It is straightforward to check, using eq.(15) , that the coefficient of a single letter in the last expression equals the coefficient of the corresponding monomial in P ˝Q.
Theorem 3.18.Let P : R d Ñ R e be a polynomial with vanishing constant coefficient.For all w P HpR e q, the relation xISS P pincq pxq n,m , wy " xISSpxq n,m , Φ P pwqy holds.In particular, ISS P pincq pxq is a quasi-shuffle character of H qsh pR e q.
Proof.We first prove the identity on SpBq." xISS P pincq pxq n,m , a 1 ¨¨¨a p y.
The quasi-shuffle property follows immediately from Lemma 3.16 .
Example 3.19.In d " 2, consider the real-valued polynomial map P px, yq " x 2 `y2 " }px, yq} 2 .This map can be of interest in a data science context, where one is interested only in the distance of the data points to the origin instead of their absolute position in the plane, e.g. if the problem has some rotational invariance properties.We then get the map (with e " 1) p ˛p1q " r1 2 s `r2 2 s.Finally, where in the last equality we have used eq.( 17) .
Remark 3.20.Even if d " e, the map Φ P is, in general, not invertible.This is due to the fact that P ´1 will in general be a formal power series and not just a polynomial.As an example, consider the single-variable polynomial P pxq " x `x3 .Its inverse satisfies P ´1pxq " x ´x3 `3x 5 ´12x 7 `opx 7 q but does not admit a finite series representation.This means that recovering ISSpxq from ISS P pincq pxq by performing a finite number of operations is not always possible.In particular, to recover the increment xISSpxq, r1sy " x N ´x0 from knowledge of ISS P pincq pxq one needs to evaluate the series xISS P pincq pxq 0,N , r1sy ´3xISS P pincq pxq 0,N , r1 3 sy `3xISS P pincq pxq 0,N , r1 5 sy ´12xISS P pincq pxq 0,N , r1 7 sy `¨¨ẅ hich might not converge depending on the size of the increment x N ´x0 .We can say that there is a "loss of information", in terms of time-warping invariance, if we are only allowed to observe some polynomial transformation of the increments of the data instead of the increments themselves.

Polynomial transformation of the time series.
We now consider polynomial transformations of the time series itself.Let P : R d Ñ R e be a polynomial map, for some e ě 1.We write P " pp 1 , . . ., p e q where p k P Rrx p1q , . . ., x pdq s is a multivariate polynomial.
Recall from Section 2 that the quasi-shuffle algebra H qsh carries a commutative quasishuffle structure.Moreover, by Theorem 2.5 it realizes the free commutative quasi-shuffle over R d ; in other words, if H is any other commutative quasi-shuffle algebra and Λ : A Ñ H is a map, there exists a unique extension Λ : H qsh Ñ H respecting the corresponding quasishuffle structures.
Given a time series x, we consider its transform X " P pxq -pP px 0 q, . . ., P px N qq, which is an R e -valued time series.Interestingly enough, the iterated-sums signature of X can be computed just by knowing that of the untransformed data x.More precisely we have (cf.[ CP20 , Theorem 2]) Theorem 3.21.Let P : R d Ñ R e , be a polynomial map without constant term, i.e., with P p0q " 0. Given a d-dimensional time series x with x 0 " 0, define the e-dimensional time series X -P pxq.Then, for all 0 ď k ď N , w P H qsh .xISSpXq 0,k , wy " xISSpxq 0,k , Λ P pwqy, where Λ P : H qsh pR e q Ñ H qsh pR d q is the unique quasi-shuffle morphism (in the sense of Definition 2.2 ), determined by its action on r1s, . . ., res as where on the righthand side, ι : Rrx p1q , . . ., x pdq s Ñ H qsh pR d q is the unique morphism of commutative algebras satisfying ιpx piq q " ris.
By adding a global shift in the variables, one can also consider polynomials with non-zero constant coefficients (cf.[ CP20 , Corollary 1]).Corollary 3.24.Let P : R d Ñ R e be a polynomial map, and x a d-dimensional time series.Define the e-dimensional time series X -P pxq, and set Px0 -P p¨`x 0 q ´P px 0 q.Then, for all 0 ď k ď N , w P H qsh .xISSpXq 0,k , wy " xISSpxq 0,k , Λ Px 0 pwqy, Proof.The result follows from Theorem 3.21 and the fact that, if x " x ¨´x 0 then x0 " 0 and ISSpxq 0,k " ISSpxq 0,k for all 0 ď k ď N .

Conclusion / Outlook
We have investigated three ways of transforming the iterated-sums signature of a time series.Using a formal power series f , the iterative definition of the signature is modified in Definition 3.7 , to obtain a generalized signature.Its relation to [ KO19 ] is sketched in Section 3.1.1 .The transformation can be realized, on the dual side, via a Hopf algebra morphism Ψ f , Proposition 3.9 .Given a polynomial P , directly transforming the increments of the time series leads to the signature ISS P pincq , ( 14 ).The transformation can be realized, on the dual side, via a Hopf algebra morphism Φ P , Theorem 3.18 .Transforming the time series itself via a polynomial P and calculating its usual signature can also be realized via a Hopf algebra morphism Φ P , Theorem 3.21 .
A remark is in order regarding the three transformations, Ψ f , Φ P and Λ P , and their properties.Regarding coalgebra morphisms, the results in [ FPT16 ] show that they only form a subset of all possible such morphisms, see Remark 2.6 and Remark 2.11 .On the other hand, we have seen that they are quasi-shuffle morphisms (in the sense of Definition 2.2 ), which implies that they satisfy the equations analog to those in Proposition 3.5 .However, it is worth noting that not all algebra morphisms are necessarily quasi-shuffle morphisms.Indeed, starting from the shuffle Hopf algebra H ¡ " pT pV q, ¡,∆ dec , ε, ηq, where T pV q is the usual unital tensor algebra over V and deconcatenation as comultiplication.We consider Φ : H ¡ Ñ H ¡ defined by Φ " e `mcon pf b Φq∆ dec , ( where e :" η ˝ε and the linear map f sends the empty word to zero, f p1q " 0, and any element of T `pV q :" À ną0 T n pV q to T 1 pV q.Here, m con denotes the concatenation product on T pV q.Following [ FPT16 , p.214], Φ is a coalgebra morphism.For an explicit proof via induction, one can use the fact that pT pV q, m con , ∆ dec , ε, ηq forms an unital infinitesimal bialgebra [ Foi09 ] characterized by the identity ∆ dec pw 1 w 2 q " pw 1 b idq∆ dec pw 2 q `∆dec pw 1 qpid b w 2 q ´w1 b w 2 , (20) for words w 1 , w 2 P T `pV q.If we further assume that f p1q " 0 " f pw 1 ¡ w 2 q for words w 1 , w 2 P T `pV q, then ( 19 ) defines a shuffle algebra morphism, i.e., for w 1 , w 2 P T `pV q, Φpw 1 ¡ w 2 q " Φpw 1 q ¡ Φpw 2 q.Note, however, that such a Φ is not a shuffle, or Zinbiel, morphism Φpxyq " Φpx ă yq " f pxyq `f pxqf pyq, which is different from Φpxq ă Φpyq " f pxqf pyq.We can naturally extend the shuffle algebra morphism Φ to the quasi-shuffle Hopf algebra using Hoffman's exponential, which gives a quasi-shuffle algebra morphism on H qsh , but not a quasi-shuffle morphism (in the sense of Definition 2.2 ).If we let ISS Φ " ISS ˝Φ be the "transformed" signature, this results on ISS Φ not satisfying a version Proposition 3.5 .We mention briefly, postponing the presentation of details to another paper, that transforming ISS to ISS Φ amounts to translations in the sense of [ BCFP19 ].Proof.This follows from the fact that for any word w and any I P Cp pwqq (defined in Section 2.2 ) we have Φ P pIrwsq " IrΦ P pwqs.
The polynomial transformation of the time series' increments and the polynomial transformation of the time series are related as follows.
Proposition 4.2.We observe that for any polynomial P with P p0q " 0, the map P ˛: SpBq Ñ SpAq of Section 3.2 can be recovered from Λ P by post-composing with the projection map π : H Ñ SpAq.That is P ˛" π ˝ΛP .

4. 1 .
Comparison of the three maps.We lastly consider now transformation maps.The following is a manifestation of the nonlinear Schur-Weyl duality (see [ FPT16 , Proposition 1.3]).Proposition 4.1.For f P tRrrtss a formal diffeomorphism and P : R d Ñ R e a polynomial with P p0q " 0, Ψ f ˝ΦP " Φ P ˝Ψf .
However, in the last sum the only word of length 1 of the form Irws is pmqrws " ra 1 ¨¨¨a m s.Let x be a d-dimensional time series and fix 0 ď n ď m ď N .The identitiesxISS f pxq n,m , u ą f vy " ÿ năkďm xISS f pxq n,k´1 , uy δxISS f pxq n,¨, vy k xISS f pxq n,m , u ‚ f vy " ÿ năkďm δxISS f pxq n,¨, uy k δxISS f pxq n,¨, vy khold for all u, v P H qsh .Taking up on Remark 3.6 , the identities contained in Corollary 3.11 also imply an efficient algorithm for the computation of ISS f .
aPA δx a n xa, Ψ f pwqy.Moreover, since the identity is linear in w, we can further restrict ourselves to the casew P A ra1¨¨¨ams n we obtain that, if w " a 1 ¨¨¨a m P A JPCpmq c i1 ¨¨¨c i k xa, Irwsy.Corollary 3.10.The generalized iterated-sums signature is a character over the Hopf algebra H f with deformed quasi-shuffle product.Also, since Ψ f is a quasi-shuffle morphism, from Proposition 3.9 we obtain the following analogue of Proposition 3.5 : Corollary 3.11." xISS f pxq n,m , r3sr5s `r35sy " xISSpxq n,m , Ψ f pr3sr5sqy.
For this, we first show it for i P B. Observe that xISSpxq n,m , p ˛piqy " If now a " ri 1 ¨¨¨i p s P B we see that xISSpxq n,m , Φ P paqy " xISSpxq n,m , rp ˛pi 1 q ¨¨¨p ˛pi p qsy " ÿ năjďm p i1 pδx j q ¨¨¨p ip pδx j q " xISS P pincq pxq n,m , ay.By linearity, the identity holds for all a P SpBq.Finally, by definition, if w " a 1 ¨¨¨a p P B ˚then xISSpxq n,m , Φ P pwqy " xISSpxq n,m , Φ P pa 1 ¨¨¨a p´1 qΦ P pa p qy " ÿ năjďm xISSpxq n,j´1 , Φ P pa 1 ¨¨¨a p´1 qyP pδx j q ap " ÿ năjďm xISS P pincq pxq n,j´1 , a 1 ¨¨¨a p´1 yP pδx j q ap